We give new rounding schemes for SDP relaxations for the problems of maximizing cubic polynomials over the unit sphere and the $n$-dimensional hypercube. In both cases, the resulting algorithms yield a $O(\sqrt{n/k})$ multiplicative approximation in $2^{O(k)} \text{poly}(n)$ time. In particular, we obtain a $O(\sqrt{n/\log n})$ approximation in polynomial time. For the unit sphere, this improves on the rounding algorithms of Bhattiprolu et. al. [BGG+17] that need quasi-polynomial time to obtain a similar approximation guarantee. Over the $n$-dimensional hypercube, our results match the guarantee of a search algorithm of Khot and Naor [KN08] that obtains a similar approximation ratio via techniques from convex geometry. Unlike their method, our algorithm obtains an upper bound on the integrality gap of SDP relaxations for the problem and as a result, also yields a certificate on the optimum value of the input instance. Our results naturally generalize to homogeneous polynomials of higher degree and imply improved algorithms for approximating satisfiable instances of Max-3SAT. Our main motivation is the stark lack of rounding techniques for SDP relaxations of higher degree polynomial optimization in sharp contrast to a rich theory of SDP roundings for the quadratic case. Our rounding algorithms introduce two new ideas: 1) a new polynomial reweighting based method to round sum-of-squares relaxations of higher degree polynomial maximization problems, and 2) a general technique to compress such relaxations down to substantially smaller SDPs by relying on an explicit construction of certain hitting sets. We hope that our work will inspire improved rounding algorithms for polynomial optimization and related problems.

翻译：我们针对单位球面与$n$维超立方体上的三次多项式最大化问题，提出了新的SDP松弛舍入方案。在这两种情形下，所得算法以$2^{O(k)} \text{poly}(n)$的时间复杂度实现$O(\sqrt{n/k})$的乘法近似。特别地，我们在多项式时间内获得了$O(\sqrt{n/\log n})$的近似保证。对于单位球面，这改进了Bhattiprolu等人[BGG+17]需拟多项式时间才能达到类似近似保证的舍入算法。在$n$维超立方体上，我们的结果匹配了Khot与Naor [KN08]通过凸几何技术获得近似比的搜索算法保证。与他们的方法不同，我们的算法能给出问题SDP松弛的整性间隙上界，从而对输入实例的最优值提供可证上界（注：原文"certificate on the optimum value"指提供最优值的认证性上界）。这些结果自然推广至更高次齐次多项式，并改进了Max-3SAT可满足实例的近似算法。我们的核心动机源于：尽管二次型SDP舍入理论已高度完善，高阶多项式优化的SDP松弛舍入技术却严重匮乏。我们提出的舍入算法包含两项创新：1）基于多项式重加权的新方法，用于高阶多项式最大化问题的平方和（SoS）松弛舍入；2）通过显式构造特定命中集，将此类松弛压缩为更小规模SDP的通用技术。我们期望这项工作能推动多项式优化及相关问题的改进舍入算法研究。